2 Answers
2

The network sites information API request includes the site logos in its response. The logos' current URL format seems to be http://sstatic.net/SITEDOMAIN/img/logo.png, where SITEDOMAIN is one of stackoverflow, apple, gamedev, etc..

For favicons, the format is http://example.stackexchange/favicon.ico and http://cdn.sstatic.net/SITEDOMAN/img/favicon.ico

For convenience, here are all of the logos by site, excluding generic betas due to the character limit: